Solution (source code)

= Solution

An economic threshold requires all unit costs to have an explicit mass basis. Let $p_U$ be the concentrate price per kilogram of <uranium>, and let $C_{\rm net}$ denote the annual incremental fabrication-and-disposal cost of the closed cycle relative to the open cycle. The break-even annual cash balance is
$$
p_U(41342.2969)+50(1486.7266)=1000M_{\rm rep}+C_{\rm net},
$$
where $M_{\rm rep}$ is the charged <nuclear fuel reprocessing> throughput in kilograms of initial heavy metal. Use $M_{\rm rep}=35393.9394$: the stated $33$ tonnes is the remaining <uranium> inventory, not the whole initial heavy-metal throughput. Thus
$$
\boxed{p_U=\frac{35393939.4+C_{\rm net}-74336.3}{41342.2969} \mathrm{dollars\,kg_U^{-1}}}.
$$
If the quoted combined $400$ dollars per kilogram is a \b[net additional charge on that same reprocessed-HM basis], $C_{\rm net}=400M_{\rm rep}$, giving \b[1,196.77 dollars per kg U]. Under this explicit convention, the saving in purchased separative work is small and the resource saving must pay nearly all the additional <nuclear fuel reprocessing> cost.

The wording does not identify the mass basis or counterfactual sign of the combined recycled-fuel and spent-fuel-disposal term. If instead $400$ dollars is a net charge per kilogram of the $4469.4375$ kg recycled product, the same balance gives \b[897.56 dollars per kg U]. If it denotes a net avoided charge of $400$ dollars per kilogram of discharged HM, the threshold is \b[511.87 dollars per kg U]. These are alternative accounting conventions, not three simultaneous economic thresholds; the general balance makes the ambiguity visible.

All prices here are per <uranium> mass, not per gross concentrate mass. This simplified comparison assigns no <plutonium> credit, financing, conversion-cost difference or change in repository liability, and takes the quoted charges as incremental. A closed fuel cycle can save natural <uranium> without being the cheaper option: at a price below the threshold under the chosen cost convention, the additional charges exceed the resource credit. No contemporary uranium-market price is needed for that conclusion.